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# Midsegment of a Triangle

A triangle is a very important and a most commonly used geometrical figure. It has three sides making an enclosed two-dimensional shape. There are various concepts and theorems based on triangles. The mid segment of a triangle plays a vital role in many theorems and applications of triangles. We are going to learn about a mid segment in this article.

When we join the middle point of a line with the middle point of another line of a triangle, the line which joins two mid points of a triangle, is known as mid segment. In other words, the mid segment of a triangle is the line segment that connects are midpoints of the two sides. In geometry, mid segment of a triangle is very important concept for solving several problems. One very useful theorem is known as mid segment theorem.

According to this theorem:

The length of the mid segment is always half of the third side of the triangle and is also parallel to the third line of the triangle.

Mid segment is always lies inside the triangle. If we know the length of all three mid segments, then we can find the length of side of bigger triangle, and also can find the perimeter and area of the triangle.

Have a look at the following diagram.

In this figure, we can say that D and E are the midpoint of Line AB and AC. So, the line DE is the mid segment of triangle ABC. Let us go ahead and learn more about mid segment of a triangle and its applications.

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## Midsegment of a Triangle Property

When we join the mid point of two line of a triangle and make a midsegment, this midsegment is parallel to the third line of triangle. (From the above figure.)

DE is parallel to BC

or  DE || BC

The area of a triangle ABC is four time the area of the midsegment of that triangle

Area of $\triangle$ ABC = 4 * (area of $\triangle$ DEF).

When we join two midpoint of two line of a triangle, then the midsegment is equal to half of its third line.

$\frac{BC}{2}$ = DE

So,  BC = 2 DE.

A triangle has maximum three midsegment.

In triangle ABC, DE, DF and EF are the midsegments.

## Midsegment of a Triangle Examples

Given below are some of the examples on midsegment of a triangle.

### Solved Examples

Question 1:

Midsegment of a triangle is DE. Find the length of third side of the triangle?

Solution:

The length of the midsegment DE is 8m

From the property of midsegment of a triangle, we know that the third side of a the triangle is double of midsegment.

DE = 8m (given)

AB = 2 $\times$ DE

AB = 2 $\times$ 8m

AB = 16 m.

Question 2:

The Midsegment of a triangle given DE, DF, FE are the length of a midsegment coresponding values are 8, 3 and 7. Find the perimeter of triangle ABC?

Solution:

The given length of midsegment of DE, DF, FE are 8, 3 and 7.

From the property of mid segment of a triangle, we know that

The third side of a the triangle is double of midsegment

If DE = 8, then AB = 16

If DF = 3, then CB = 6

If FE = 7, then AC = 14

The perimeter of $\triangle$ ABC is (6 + 14 + 16)

Thus, the perimeter of $\triangle$ ABC is 36.

 More topics in Midsegment of a Triangle Triangle Midsegment Theorem Side Splitter Theorem
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